PSI - Issue 2_A

Stefano Bennati et al. / Procedia Structural Integrity 2 (2016) 2682–2689 S. Bennati, D. Colonna and P.S. Valvo / Structural Integrity Procedia 00 (2016) 000–000

2686

5

In a real structure, cambering of the beam is often introduced to compensate for the deflection due to dead loads. For simplicity, here we do not consider this and assume the deformed configuration of the unstrengthened beam as the reference configuration for the strains and displacements calculated below (from stage 1 on).

3.2. Stage 1 – Pre-stressing and fixing of the laminate

During the pre-stressing stage, the laminate is put into tension by an axial force, P , applied through the anchor points on the beam bottom surface. Simultaneously, the beam is compressed by the same axial force, which produces also bending because of the eccentricity of the load application point with respect to the beam centreline. The internal forces produced by pre-stressing in the reinforced part of the beam ( 0 ≤ ≤ s l ) are

1 2

, b P N s

, b P P V s ,

, b P M s

Ph

= −

( ) 0, =

= −

( )

( )

.

(3)

b

The internal forces given by Eqs. (3) are to be added to those given by Eqs. (2) to obtain the total internal forces in the beam. All loads should be suitable factored when evaluating ultimate limit states. Due to pre-stressing, points belonging to the beam bottom surface will move towards the mid-span cross section; conversely, points on the laminate will move towards the anchor point C . From the classic assumption of beam theory, that plane sections remain plane, and the constitutive laws for the beam and laminate, we determine the following (positive) displacement for a point S placed on the beam bottom surface at the abscissa s :

  

  

2

h

1

(

)

, b P w s P ( ) =

− l s

+

;

b

(4)

4 E A E I s b s b

and the following (negative) displacement for a point S * of the laminate at the abscissa s *:

P

(

) * .

f P w s

− l s

= −

, ( *)

(5)

E A

f

f

On curing of the adhesive, the beam and laminate behave as a composite structure. To determine the interfacial stresses through Eq. (1), the relative displacements at the interface must be evaluated with respect to the deformed configuration at the end of the pre-stressing stage. To this aim, we consider that points S and S *, initially not aligned, be placed on the same cross section at the end of the pre-stressing operation (Fig. 5).

Fig. 5. Displacements of beam and laminate at the end of the pre-stressing stage.

Alignment of points S and S * after pre-stressing requires that

(6)

, b P f P s w s s w s + = + , ( ) *

( *).